Question: Multiply the following complex numbers: $({5+i}) \cdot ({1+4i})$
Solution: Complex numbers are multiplied like any two binomials. First use the distributive property: $ ({5+i}) \cdot ({1+4i}) = $ $ ({5} \cdot {1}) + ({5} \cdot {4}i) + ({1}i \cdot {1}) + ({1}i \cdot {4}i) $ Then simplify the terms: $ (5) + (20i) + (1i) + (4 \cdot i^2) $ Imaginary unit multiples can be grouped together. $ 5 + (20 + 1)i + 4i^2 $ After we plug in $i^2 = -1$ , the result becomes $ 5 + (20 + 1)i - 4 $ The result is simplified: $ (5 - 4) + (21i) = 1+21i $